Definition Of Limits

Now that we have looked at how limits arise when we calculate the area of a curved
figure let us investigate the numerical and graphical methods of computing limits.

Consider the function f (x) = x
2 + x + 1

We will take a look at how f (x) behaves for values of near 2. The following
table will give the values of f (x) for near 2 but not equal to 2.

x

f (x)

1

3

1.5

4.75

1.7

5.59

1.9

6.51

1.99

6.9501

1.999

6.995001

x

f (x)

3

13

2.5

9.75

2.2

8.04

2.1

7.51

2.01

7.0501

2.001

7.005001

From the graph and the table we can see that as approaches 2 from the left, f (x)
gets closer to 7. Similarly as approaches 2 from the right f (x) approaches
7. We can further verify that f (x) can be made as close to 7 as required
by taking sufficiently closer to 2.

In general we can say that
limx2+
x + 1 = 7
x->2

Definition: Symbolically, we write limf (x)
= L and read this as
x->a

"The limit of f (x) as x approaches a is L"

If f (x) can be made arbitrarily close to L by taking x sufficiently
close to a and x ≠ a.

We can also write this as f (x) -> L as x -> a.

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